论文标题
关于巴拉克空间上的halmos的第三个问题
On the third problem of Halmos on Banach spaces
论文作者
论文摘要
假设$ x $是一个复杂的可分开的无限尺寸Banach空间，$ \ Mathcal {b}（x）$表示所有有界线性运算符的Banach代数从$ x $到本身。 1970年，P.R。Halmos在希尔伯特空间中提出了十个开放问题。第三个是以下内容：如果不及物的操作员$ t $具有逆，它的逆也不超过吗？这个问题与不变的子空间问题密切相关。自从Enflo在$ \ ell_1上庆祝的反例以来，否定了不变的子空间问题，Halmos第三个问题的Banach空间设置变得更加有趣。在本文中，我们在某些光谱条件下对此问题给出了肯定的答案。作为一个应用程序，我们表明，对于可抗可逆的操作员$ t $带有邓福德的财产（$ c $），如果$ t^{ - 1} $是不及物的，并且存在连接的组件$ω$ of $intσ（t^{ - 1}）^\ land $，则是$ capept $ neq $ $ capept $ ne $ neq $ neq $ ne $ 1}。也不及物。在本文的最后，我们表明存在一个足够且必要的条件，存在一个有界的线性操作员，而无需非平凡的不变子空间在无限尺寸空间上$ l_1（ω，\ sum，μ）$（resp。，$ c（k）$，$ k $ k $ $ k $）是（$ c（k）$（$ c（k）$）， $σ$ -finite（resp。，$ k $是紧凑的）。
Assume that $X$ is a complex separable infinite dimensional Banach space and $\mathcal{B}(X)$ denotes the Banach algebra of all bounded linear operators from $X$ to itself. In 1970, P.R. Halmos raised ten open problems in Hilbert spaces. The third one is the following: If an intransitive operator $T$ has an inverse, is its inverse also intransitive? This question is closely related to the invariant subspace problem. Ever since Enflo's celebrated counterexample on $\ell_1$ answered the invariant subspace problem in negative, the Banach space setting of the third question of Halmos has become more interesting. In this paper, we give an affirmative answer to this problem under certain spectral conditions. As an application, we show that for an invertible operator $T$ with Dunford's Property ($C$), if $T^{-1}$ is intransitive and there exists a connected component $Ω$ of $intσ(T^{-1})^\land$ which is off the origin such that $Ω\capρ_F(T^{-1})\neq \emptyset$, then $T$ is also intransitive. In the end of the paper, we show that a sufficient and necessary condition for that there exists a bounded linear operator without non-trivial invariant subspaces on the infinite dimensional space $L_1(Ω,\sum,μ)$ (resp., $C(K)$, the space of bounded continuous functions on a complete metric space $K$) is that $(Ω,\sum,μ)$ is $σ$-finite (resp., $K$ is compact).
